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Surface Area of a Cone

Date: 06/18/98 at 18:22:31
From: Arthur Schultz
Subject: Surface area of a cone

What is the formula for the surface area of a cone?

I have looked through several sites, but everyone seems to be obsessed 
with volumes.

Thanks for your help,
Arthur Schultz

Date: 06/26/98 at 18:04:09
From: Doctor Floor
Subject: Re: Surface area of a cone

Hi Arthur,

Thank you for sending your question to Dr. Math. I hope to be of some 

I will try to show you how to compute the surface area of (the curved 
part of) a cone for which the height and top angle are given.

To compute the area of the surface of a cone you should think of how 
to construct a cone from a piece of paper: First you make a circle, 
then you cut an angle out from its midpoint, and after that you bring 
the two angle legs of the remaining part together.

From this you see that the surface area of (the curved part of) the 
cone is a part of a circle. But still we have to find out how to 
compute the radius of this circle and the angle of the part to 
construct it from. So let's look at a vertical cut of a cone:

              / | \        ATC = alpha (given)
             /  |  \         h = height (given)
           r/  h|   \r       r = radius of circle to construct cone from
           /    |    \      r1 = radius of bottom circle
          /     |     \
         A -----B-----C
            r1     r1

In the picture you see that angle 

  ATB = alpha/2

so r can be computed using 

  cos(alpha/2) = h/r

             r = h/cos(alpha/2).

Now I want to emphasize the importance of knowing about r1, the radius 
of the bottom circle, compared to r:

Let's say that you construct a cone by cutting out 1/4 of the circle 
with radius r, so that you use 3/4 of the circle to construct the cone. 
Then the original circumference equals 2*pi*r, and the circumference 
of the bottom circle becomes 3/4 of the original, or (3/4)*2*pi*r. 

In this way the radius of the bottom circle r1 is 

  r1 = ((3/4)*2*pi*r)/(2*pi) 

     = (3/4)*r

We can conclude that r1/r = 3/4 equals the part of the original circle 
that was needed to construct the cone. Since 3/4 can be replaced in the computation by any other number, with the same result that it is equal 
to r1/r, we can conclude that the part of the original circle needed is 
r1/r in all cases.

From the figure we can see that 

  r1/r = sin(alpha/2).


    The surface area of the cone 
    = r1/r * area of the "original circle"
    = sin(alpha/2) * area of the "original circle"

    = sin(alpha/2) * pi * r^2
    = sin(alpha/2) * pi * (h/cos(alpha/2))^2

    = pi * h^2 * sin(alpha/2) / cos^2(alpha/2)

I hope this is all clear!

You might also like to look at the Geometric Formulas area of the Dr. 
Math FAQ:   

Best regards,

- Doctors Floor and Sarah, The Math Forum   
Associated Topics:
High School Geometry
High School Higher-Dimensional Geometry
High School Trigonometry

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